Axial Vibration Control Technique for Crystal Growth from the Melt: Analysis of Vibrational Flows’ Behavior

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors investigated the axial vibrational control for crystallization from solution or melt by using numerical simulation and determine the analytical dependence of the maximum melt velocity on the system parameters including amplitude, frequency, radius of the disk edge, density and viscosity of the melt in a closed crucible. The results are interesting and systematic, and therefore it is recommended to publish.

Author Response

Thank you for the kind review. We will try to do our best to improve the submission according to the comments of all reviewers.

Reviewer 2 Report

Comments and Suggestions for Authors

Ultrasonic vibrations are rarely used in the context of crystal growth but it is known that the vibration causes a global convection.
In this paper numerical investigations are presented using an axisymmetric approach.

This is an interesting study, because not much has been done in this area.


Model:

I am a little bit puzzled about the labeling of the coordinates> the most convenient would be to use r and z, for the radial and axial directions, respectively.
I guess in Eq (5) is the axial direction labeled with x.

I expect that Eq (5) is applied at all boundaries of the disk.
I understand that the disk is moved according to the velocity given in (5). This means that all nodes of the disk are moved by this velocity.
Thus, in the melt a poisson equation is solved to adapt the grid to the new boundaries ?


At certain points in the text it is not clear if instantaneous or time-averaged velocities are shown.
For instance in Fig. 2..  In addition the velocity vectors are too small.

In Fig. 3 the boundary layer in case of Ge i thicker than for CdTe. But the text on pg 5. says the contrary.

Eq 7 and 8.

How were the exponents derived ?
I understand I = rho^0.5 ....     and not I proportional to rho^0.5 .... as seed by Eq. 8

However,  from Fig. 6 on would guess something like v = sqrt(I)

only for small I it seems to be linear.   Is this a saturation effect.

In any case a more careful description is needed.

pg 9 bottom line
Do the authors mean the "mean" velocity or the "maximum" velocity ?

So far I understand that the authors used the maximum velocity for the equations as shown in Table 3.

Turbulence etc.

There is the interesting question when the melt convection becomes really time-dependent.
I mean when is the flow integrated over on period of oscillation is not longer stable.
I do not see any turbulent flow in the figures and I would be careful with this expression.
Typically, one starts with a time-dependent flow with a certain period.

Altogether, it is an interesting topic but the presentation has to be improved to make it valuable to the readers.

Author Response

Dear Reviewer

Thanks a lot for the fruitful comments. We corrected the text according to the recommendation.

I am a little bit puzzled about the labeling of the coordinates> the most convenient would be to use r and z, for the radial and axial directions, respectively.
I guess in Eq (5) is the axial direction labeled with x.

Reply
We corrected the axis’s according to cylindrical coordinate system

I expect that Eq (5) is applied at all boundaries of the disk. I understand that the disk is moved according to the velocity given in (5). This means that all nodes of the disk are moved by this velocity. Thus, in the melt a poisson equation is solved to adapt the grid to the new boundaries ?

Reply

We added the explanation of the mesh motion.

At certain points in the text it is not clear if instantaneous or time-averaged velocities are shown.
For instance in Fig. 2..  In addition the velocity vectors are too small

Reply.
We used time-averaged velocities. We changed the Figure 2 and added the new explanation/

In Fig. 3 the boundary layer in case of Ge i thicker than for CdTe. But the text on pg 5. says the contrary. Eq 7 and 8.

How were the exponents derived ?  I understand I = rho^0.5 ....     and not I proportional to rho^0.5 .... as seed by Eq. 8 However,  from Fig. 6 on would guess something like v = sqrt(I)
only for small I it seems to be linear.   Is this a saturation effect. In any case a more careful description is needed.

Turbulence etc. There is the interesting question when the melt convection becomes really time-dependent. I mean when is the flow integrated over on period of oscillation is not longer stable.
I do not see any turbulent flow in the figures and I would be careful with this expression.
Typically, one starts with a time-dependent flow with a certain period.

Reply.
Agreed. We added the figures and corrected the main text based on the reviewer comments.

pg 9 bottom line
Do the authors mean the "mean" velocity or the "maximum" velocity ? So far I understand that the authors used the maximum velocity for the equations as shown in Table 3.

Reply.
We corrected the phrase.

“Numerical simulation showed that the maximum (averaged over the oscillation period) velocity of vibrational flows was inversely to R and demonstrated a non-linear increase with AVC amplitude (A).”

Reviewer 3 Report

Comments and Suggestions for Authors

To author:

1.The innovation of the article is not clear. Compared with the existing schemes of the same type, what is the difference in the innovation of this article? The literatures review in the introduction are not sufficient, the readability and presentation need to be further improved.

2.The author mentions in lines 65-69 on page 2 that in order to determine the direct relationship between the parameters of vibration effects and the characteristics of vibration flows in a liquid, they excluded all factors. It would be helpful to provide guidance on how to validate experimental results when there are interferences in practical situations, as suggested by the author's research approach.

3.In lines 101-102 on page 3, supplementary equations are introduced. It is recommended to provide explanations for the parameters used in these supplementary equations.

4.Is the theory proposed in this paper applicable to the preparation of semiconductor silicon single crystals, or is it only applicable to Ge, NaNO3, CdTe as mentioned in the paper?

5.What are the limitations of the method proposed in this paper? Additionally, it would be beneficial to suggest future research directions.

Comments on the Quality of English Language

1.The paper should be checked by a native.

Author Response

Dear Reviewer,

Thanks a lot for the fruitful comments. We added the necessary explanation according to your comments and we believe that this strongly enhanced the article 

1. The innovation of the article is not clear. Compared with the existing schemes of the same type, what is the difference in the innovation of this article? The literatures review in the introduction are not sufficient, the readability and presentation need to be further improved.

Replay.
We added the main goal of the research in the Introduction section

2. The author mentions in lines 65-69 on page 2 that in order to determine the direct relationship between the parameters of vibration effects and the characteristics of vibration flows in a liquid, they excluded all factors. It would be helpful to provide guidance on how to validate experimental results when there are interferences in practical situations, as suggested by the author's research approach.

Reply.
We added the phrase about further investigations in the Conclusion section.

“Further research involves conducting physical experiments in an isothermal version in order to validate the data obtained. Next, the performance of the model will be tested on non-isothermal systems, taking into account the crystallization process on low-temperature model substances. After this, the developed model can be used to optimize the growth processes of high-temperature crystals.”

3.In lines 101-102 on page 3, supplementary equations are introduced. It is recommended to provide explanations for the parameters used in these supplementary equations.

Replay.
Agreed. Added the explanation of the parameters

4. Is the theory proposed in this paper applicable to the preparation of semiconductor silicon single crystals, or is it only applicable to Ge, NaNO3, CdTe as mentioned in the paper?

Replay.
It was our main goal the expand the AVC technique on other actual materials including semiconductors.

5. What are the limitations of the method proposed in this paper? Additionally, it would be beneficial to suggest future research directions.

Replay.
In conclusion, it should be noted that the AVC method is very energy efficient in terms of controlling flows in the liquid phase [25, 26]. At the same time, the AVC method allows one to potentially overcome the scaling problem when increasing the diameter of the growth system from 20 mm to 200 mm or more without significantly changing the design of the vibration actuator.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Changes and extensions have been made and the paper can be published.

Author Response

Thank you for the fruitful comments which allows us to enhance the article

Reviewer 3 Report

Comments and Suggestions for Authors

As mentioned earlier, the author stated on page 2, lines 65-69, that in order to determine the direct relationship between the parameters of vibration effects and the characteristics of vibration flows in a liquid, they excluded all factors. When answering this question, the author only added plans for further research in the conclusion, but did not actually address the issue. After excluding factors in the objective environment, can the established mechanism model represent the actual process of crystal growth?

Comments on the Quality of English Language

The quality of the English language is generally satisfactory.

Author Response

Thank you for the fruitful comments which allows us to enhance the article

We added the explanation in the Conclusion section

When considering thermoconvective flow, for example, in the VGF method, without taking into account the crystallization heat of the substance, we can assume that there will be practically no flow motion in the melt due to the fact that the hotter melt layer will be located above the colder one. In this case, the side heater creates an axial temperature gradient, which leads to the melt movement from bottom to top along the wall. Taking into account the crystallization heat of the substance, the melt will be set in motion. Depending on the crystal growth rate, the crystallization heat value, and the thermal conductivity of a given substance in the melt and solid, thermal convection in the absence of vibrations can be easily assessed [26].

The general analysis we performed in this work let us give an answer to what vibration parameters must be set in order to form a vibrational melt flow capable of leveling the crystallization front for substances with different physico-chemical and thermodynamic parameters.

26. Nefedov, O.; Dovnarovich, A.; Kostikov, V.; Mozhevitina, E.; Bocharnikov, D.; Avetissov, I. Numerical Simulation of CdTe Crystal Growth Using the Vertical Gradient Freeze Technique Assisted by Axial Low-Frequency Oscillations of the Melt. Crystals 2024, 14 (1), 72. https://doi.org/10.3390/cryst14010072 .